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Interesting connections and analogies have been observed between non-linear geometry of Banach spaces and coarse geometry. In the former subject, people have investigated the notion of uniform (or Lipschitz) quotient maps. See, e.g., this paper. The corresponding notion for coarse geometry is coarse quotient maps. We say a map $q\colon X\to Y$ between metric spaces is a coarse quotient map if $\exists K>0$ $\forall R>0$ $\exists S>0$ such that for all $x\in X$ one has $q(B(x,R)) \subset B(q(x),S)$ and $B(q(x),R) \subset N_K(q(B(x,S)))$. Here $B$ denotes the ball and $N_K$ denotes the $K$-neighborhood.

When $G$ is a countable discrete group with a proper left invariant metric, every quasi-morphism $q\colon G\to {\mathbb R}$ is a coarse quotient map. In fact, I view a coarse quotient map from $G$ to ${\mathbb R}$ as a metric analogue of a quasi-morphism.

I wonder if coarse quotient maps are worth studying.

Here's a sample question: Does $\mathrm{SL}(3,\mathbb{Z})$ admit a coarse quotient map to ${\mathbb R}$?

Added: This notion has been studied in Sheng Zhang Israel Journal of Mathematics volume 207, pages 961-979 (2015) (ArXiv, DOI link, MR link).

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    $\begingroup$ A remark. Clearly a map $X\to Y$ satisfying $q(B_X(x,R))=B_Y(q(x),R)$ satisfies the condition, and composing with QIs preserve the conditions. In particular, for every infinite finitely generated group with word length $|\cdot|$, the map $g\mapsto |g|$ is a "coarse quotient map" $G\to \mathbf{R}_{\ge 0}$. $\endgroup$
    – YCor
    Commented Sep 15, 2021 at 8:27
  • $\begingroup$ (Sorry my last comment was senseless, I erased it) $\endgroup$
    – YCor
    Commented Sep 16, 2021 at 7:06
  • $\begingroup$ And my first (non-erased) comment was too hasty too: there might be "dead ends" in a finitely generated group and the the condition can't fail. However if a group is QI to a metric space in which any two points belong to a bi-infinite geodesic, it works. This applies at least to cocompact lattices in $\mathrm{SL}_n(\mathbf{R})$ for $n\ge 2$. $\endgroup$
    – YCor
    Commented Sep 16, 2021 at 9:33

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